Chapter 12: Q. 9 (page 989)

Evaluate the following limits, or explain why the limit does not exist.
$\lim_{(x, y, z) \to (1, 0, - 1)} \frac{\sin (x y)}{x^{2} - y^{2} + z^{2}}$

Short Answer

Expert verified

$\lim_{(x, y, z) \to (1, 0, - 1)} \frac{\sin (x y)}{x^{2} - y^{2} + z^{2}} = 0$

Step by step solution

Step 1. Given information is:

$\lim_{(x, y, z) \to (1, 0, - 1)} \frac{\sin (xy)}{x^{2} - y^{2} + z^{2}}$

Step 2. Evaluating Limits

$Since, (x, y, z) \to (1, 0, - 1) \sin (xy) \to 0 and x^{2} - y^{2} + z^{2} \to 2 which is finite So the function \frac{\sin (xy)}{x^{2} - y^{2} + z^{2}} is continuous at (1, 0, - 1) . So, \lim_{(x, y, z) \to (1, 0, - 1)} \frac{\sin (xy)}{x^{2} - y^{2} + z^{2}} = \frac{\sin (0)}{1^{2} - 0^{2} + {(- 1)}^{2}} = 0 Hence, \lim_{(x, y, z) \to (1, 0, - 1)} \frac{\sin (xy)}{x^{2} - y^{2} + z^{2}} = 0$

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Evaluate the following limits, or explain why the limit does not exist.
$\lim_{(x, y, z) \to (1, 0, - 1)} \frac{\sin (x y)}{x^{2} - y^{2} + z^{2}}$

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Evaluating Limits

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Evaluate the following limits, or explain why the limit does not exist.lim(x,y,z)→(1,0,-1)sin(xy)x2-y2+z2

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Evaluating Limits

One App. One Place for Learning.

Most popular questions from this chapter

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Evaluate the following limits, or explain why the limit does not exist.
$\lim_{(x, y, z) \to (1, 0, - 1)} \frac{\sin (x y)}{x^{2} - y^{2} + z^{2}}$